This paper introduces a small-uniform statistic \(W_n\)  that is constructed as a fractional programming problem out of the FPCA estimator \(\hat{\rho }\)  of the slope \(\rho \)  of the functional linear model. Our main result Theorem REF  shows \(W_n\)  converges in probability to the supremum of a Gaussian process \(\widetilde{W}_n\) . The key arguments to showing our main result are by taking advantage of identifying the regressors' underlying Hilbert space with its dual, and also recent advances by [1]} in studying the suprema of empirical processes indexed by functionals.
